Will this mesh work to rig and animate the lips?
Depends on what you mean by "rig."  If you provide proper weights and proper bones, yes, it will deform, basically how you tell it to.

If you're referring to autoweights, then it might be useful to think of weights as a fluid that can "spill" out of holes in the mesh.  Holes like this can cause undesirable weights from automatic weights.  But it's easy to fix: just face the hole (ngon is fine) before you autoweight, then delete the face afterwards.

Note that your eyes are just as much a source of this issue as your mouth.

Finally, you might mean, will this deform well?  To answer that, I'd really need to see a wireframe, but from what I can see, no it will not.  It appears that your face is a big grid without any attention paid to edge flow.  You should start looking for references on face topology.